MTG Arena Draft vs. Pack Opening

// --- results ---

{
    packRates:     {
                    wildcard: {
                                common:   0.3333333333333333,
                                uncommon: 0.3666666666666667,
                                rare:     0.16979166666666667,
                                mythic:   0.03854166666666667,
                            },
                    regular:  {
                                common:   4.666666666666667,
                                uncommon: 1.8,
                                rare:     0.8385416666666667,
                                mythic:   0.11979166666666667,
                            },
                },
    draftSim:      {
                    winExpectation:     [
                                        0.0625,
                                        0.125,
                                        0.15625,
                                        0.15625,
                                        0.13671875,
                                        0.109375,
                                        0.08203125,
                                        0.171875,
                                    ],
                    averageWins:        3.7578125,
                    averageReward:      3.6708593749999996,
                },
    fivePacks:     {
                    wildcard: {
                                common:   1.6666666666666665,
                                uncommon: 1.8333333333333335,
                                rare:     0.8489583333333334,
                                mythic:   0.19270833333333334,
                            },
                    regular:  {
                                common:   23.333333333333336,
                                uncommon: 9,
                                rare:     4.192708333333334,
                                mythic:   0.5989583333333334,
                            },
                },
    draftRewards:  {
                    wildcard: {
                                common:   1.2236197916666665,
                                uncommon: 1.3459817708333333,
                                rare:     0.6232813313802082,
                                mythic:   0.14148103841145832,
                            },
                    regular:  {
                                common:   27.13067708333333,
                                uncommon: 9.607546875,
                                rare:     3.9531685384114583,
                                mythic:   0.5647383626302083,
                            },
                },
}

// --- calculation ---

/*
    Expected outcome and reward for a draft
*/

function simulateDraftgames (matchWinrate) {
    const rewards = [1.45, 1.72, 2.24, 2.76, 3.55, 4.60, 5.65, 6.75];  // booster packs plus gems converted to booster packs
    const winExpectation = Array(8).fill(0);

    const p = matchWinrate;
    (function iterate (w, l, pathProbability) {
        if (w === 7) {
            winExpectation[w] += pathProbability;
            return;
        }
        if (l === 3) {
            winExpectation[w] += (1 - p) * pathProbability;
            iterate(w + 1, l, p * pathProbability);
            return;
        }
        iterate(w + 1, l,     p       * pathProbability);
        iterate(w,     l + 1, (1 - p) * pathProbability);
        return;
    })(0, 0, 1);

    const averageWins = winExpectation.reduce( (acc, p, w) => acc + p * w );
    const averageReward = winExpectation.reduce( (acc, p, w) => acc + p * rewards[w] );

    return {winExpectation, averageWins, averageReward};
}

/*
    Expected outcome from opening booster backs
*/

const mythicPackRate = 1/8;
const mythicWheelRate = 1/5;

const wildcards = {
    replaceChance: {
        common:   1/3,
        uncommon: 1/5,
        rare:     1/24,
        mythic:   1/24,
    },
    rewardRate: {
        common:   0,
        uncommon: 1/6,
        rare:     (1 - mythicWheelRate) * 1/6,
        mythic:   mythicWheelRate       * 1/6,  // every 5th rare wildcard reward is a mythic instead of a rare
    },
};

const packRates = {
    wildcard: {
        common:                          wildcards.replaceChance.common   + wildcards.rewardRate.common,
        uncommon:                        wildcards.replaceChance.uncommon + wildcards.rewardRate.uncommon,
        rare:     (1 - mythicPackRate) * wildcards.replaceChance.rare     + wildcards.rewardRate.rare,
        mythic:   mythicPackRate *       wildcards.replaceChance.mythic   + wildcards.rewardRate.mythic,
    },
    regular: {
        common:   5                    - wildcards.replaceChance.common,
        uncommon: 2                    - wildcards.replaceChance.uncommon,
        rare:     (1 - mythicPackRate) * (1 - wildcards.replaceChance.rare),
        mythic:   mythicPackRate       * (1 - wildcards.replaceChance.mythic),
    },
};

function expectedCardAmount (packs) {
    return {
        wildcard: Object.fromEntries(
            Object.entries(packRates.wildcard).map( ([rarity, rate]) => [rarity, packs * rate] )
        ),
        regular: Object.fromEntries(
            Object.entries(packRates.regular).map( ([rarity, rate]) => [rarity, packs * rate] )
        ),
    };
}

/*
    calculating some examples
*/

const fivePacks = expectedCardAmount(5);

const draftSim = simulateDraftgames(0.5);

const draftRewards = expectedCardAmount(draftSim.averageReward);
draftRewards.regular.common += 10;
draftRewards.regular.uncommon += 3;
draftRewards.regular.rare += 1 - mythicPackRate;
draftRewards.regular.mythic += mythicPackRate;

return {packRates, draftSim, fivePacks, draftRewards};